Abstract
Acomplete quantummechanical description of matter and its interaction with the environment requires detailed knowledge of a number of complex parameters. In particular, information about the phase of wavefunctions is important for predicting the behaviour of atoms, molecules or larger systems. In optics, information about the evolution of the phase of light in time^{1} and space^{2} is obtained by interferometry. To obtain similar information for atoms and molecules, it is vital to develop analogous techniques. Here we present an interferometric method for determining the phase variation of electronic wave packets in momentum space, and demonstrate its applicability to the fundamental process of singlephoton ionization. We use a sequence of extremeultraviolet attosecond pulses^{3,4} to ionize argon atoms and an infrared laser field, which induces a momentum shear^{5} between consecutive electron wave packets. The interferograms that result from the interaction of these wave packets provide useful information about their phase. This technique opens a promising new avenue for reconstructing the wavefunctions^{6,7} of atoms and molecules and for following the ultrafast dynamics of electronic wave packets.
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The development of intense femtosecond lasers, capable of generating electric fields with strengths comparable to the Coulomb fields in atoms and molecules, has motivated extensive fundamental research in the past twenty years. Ionization and dissociation processes are increasingly well understood, and there have been important spinoffs, such as the development of coherent light sources in the extremeultraviolet (XUV) range by highharmonic emission, in particular the generation of attosecond light pulses^{3,4}. This research field is now beginning to make important contributions in other areas of science. For example, electrons ionized by a strong laser field have been used to probe molecular structure^{8} and molecular groundstate wavefunctions^{7}. The new idea put forward in our work is to make use of interferences between continuum electron wave packets^{9} that are prepared by a sequence of attosecond laser pulses. This allows us to probe the dynamics of continuum electrons in an infrared (IR) laser field, to investigate the fundamental process of atomic photoionization and to characterize the electronic wave packet created in the process.
Our method closely resembles the traditional implementation of interferometry in optics, where a light wave E(x) is split into two replicas that travel different paths. The two replicas interfere when they are recombined, and from the interference pattern, E(x)+E(x)e^{iδϕ(x)}^{2}∝1+cosδ ϕ(x), the difference in the accumulated phase, δ ϕ(x), in the two paths can be extracted. The interference pattern in this case does not depend on the phase of the initial field, but on the optical path difference. In another implementation, the field itself can be characterized if a ‘shear’ δ x is added to one of the two replicas. If E(x)≈E(x+δ x) in the region of overlap,
where φ(x) is the phase of E(x). In this case, information about the phase φ(x) can be retrieved. This technique has been applied extensively in optics, for the measurement of wavefronts^{2} when a spatial shear is induced, and for the determination of the spectral phase of ultrashort optical pulses^{1,5} when a spectral shear is induced.
The principle of our measurement technique is illustrated, for helium, in Fig. 1. The photoelectronmomentum distribution obtained when a helium atom is ionized by a single attosecond pulse is shown in Fig. 1a. It is a ringshaped distribution peaked along the direction of polarization of the electromagnetic field^{10}, which here corresponds to the p_{y} axis. When an IR field is present at the time of ionization, the momentum distribution is shifted by the amount of momentum transferred from the field to the continuum electron wave packet, as shown in Fig. 1b. Here, the infrared intensity is I_{IR}=3×10^{13} W cm^{−2} and the momentum transfer is equal to −e A(τ)=1×10^{−24} N s, where A is the vector potential of the IR field, polarized in the same direction (y) as the XUV field, τ is the instant of ionization and e is the electron charge. This effect, which strongly depends on the initial timing^{11,12}, has previously been used in attosecond metrology to determine the characteristics of single attosecond pulses^{13} as well as of the IR laser field^{14}.
When the atom is ionized by two attosecond pulses separated by half the IR laser period, two electron wave packets are created which will interfere where their momentum distributions overlap. This is very similar to the traditional Young’s doubleslit experiment, where the slits here are provided by the doubleattosecond pulse excitation. When the electron wave packets are formed in the presence of an IR field at times when the vector potential is zero (Fig. 1c), they have approximately the same final momentum distributions but different accumulated phases due to the different amount of time spent in the dressed continuum. As explained in the Supplementary Information, using a semiclassical formulation^{15,16} for δpulses, the phase difference between the two interfering wave packets can be written as
where m is the electron mass, ħ is the Planck constant divided by 2p, ω is the IR frequency, I_{p} is the ionization energy, A_{0} is the amplitude of the vector potential, and U_{p}=e^{2}A_{0}^{2}/4m is the ponderomotive energy. The total energy absorbed in the ionization process is . As shown in Fig. 1c, the interference pattern, obtained by solving the timedependent Schrödinger equation in helium for 180attosecond pulses, follows the semiclassical predictions (indicated by the white circle) remarkably well. This shows that neither the presence of the ion core, modifying the electron dynamics, nor the nonzero length of the attosecond pulses, significantly affect the interferogram, which consists of circles centred at (p_{x}=0,p_{y}=±2e A_{0}/π). The sign depends on the direction (up or down) of the vector potential during the electron trajectory (in Fig. 1c, it is positive). The position of the centre of the circles can be used to determine the intensity of the IR field, and their radii allow us to crosscheck the value of the ponderomotive shift U_{p} (here equal to ≈1.8 eV). However, no information about the phase of the ‘initial’ continuum electron wave packet can be deduced. (‘Initial’ here and throughout the text refers to the wave packet created in the absence of an infrared laser field.)
When the attosecond pulses instead coincide with the maxima and minima of the vector potential (Fig. 1d), the momentum transfer from the IR field is maximal and opposite in direction for the two wave packets. The difference in accumulated phase between the two ionization events due to the interaction with the field is equal to zero in this case. The phase that governs the interference pattern can be written as (see the Supplementary Information)
This allows us to determine the difference between the phases of the initial electron wave packet (φ) at two different positions in momentum space, in the region where the two final momentum distributions overlap. To illustrate this point, the interference pattern inside the white rectangle in Fig. 1d is analysed (Fig. 2a). The phase can be extracted from the interferometric measurement by using a Fouriertransform method^{17}, allowing us to extract the phase of the oscillating fringes at different points in momentum space using straightforward mathematical manipulations. The most remarkable feature exhibited in the phase of the fringes, represented in Fig. 2b, is the abrupt change of phase by π at p_{y}≈±e A_{0}, which corresponds to the momentum shift induced by the IR field. In Fig. 2c, we extract . Δφ is approximately constant, equal to π for p_{y}≤e A_{0} and 0 elsewhere. This indicates that the phases of the initial wave packet at and differ by 0 or π, depending on whether the momenta are located on the same side or on opposite sides of the p_{y}=0 plane (see the schematic diagram) respectively. This result is expected for photoionization of the helium ground state, and leads to a pcontinuum wavefunction which changes sign across the p_{y}=0 plane.
More generally, our interferometric technique should allow us to visualize any phase variation of the initial continuum wave packet that is in the region of overlap of the final momentum wave packets. However, in this proofofprinciple example, the phase retrieval is somewhat limited. To improve our technique towards a morecomplete phase reconstruction, we must (1) use a momentum shear in two perpendicular directions, as in lateralshearing interferometry^{2}, to enable the reconstruction of the phase in two dimensions; (2) use a combination of small momentum shear to obtain a larger region of overlap and to better sample the momentum distribution, and large momentum shear to probe the difference of phase between the different parts of the electron wave packet.
In our experiment (Fig. 3a), a train of attosecond pulses, generated through highorder harmonic generation in argon^{18}, is used to ionize argon, resulting in a train of electron wave packets with 200±30attosecond duration, a central kinetic energy of 11 eV and a bandwidth of 11 eV. The pulse duration of the electron wave packets are determined by a RABITT (reconstruction of attosecond beating by interference of twophoton transition) analysis^{3}. The IR field, which accelerates the continuum electron wave packets, is a fraction of the laser beam used to generate the harmonics. The key results of this work are presented in Fig. 3b,c, where twodimensional cuts through the threedimensional momentum distributions^{19,20} obtained in argon using a velocitymap imaging technique^{21,22} are presented for two delays between the XUV and IR fields. The distributions are more complex than those presented in helium. This is partly because of the properties of our attosecond pulses, which are generated as a train, containing more than two pulses. An additional interference structure is superposed on that described previously, which consists of circles centred at the origin, simply expressing energy conservation. This interference effect is very similar to that leading to electron peaks separated by ħω, when atoms are ionized by intense laser pulses^{23,24}. In addition, argon has three different initial states, 3p,m=0,±1 and, for onephoton absorption, several possible final states with s or d symmetry. The interferences discussed above are, however, clearly visible. Figure 3b shows a complex interference structure, with three sets of circles that have different origins: (p_{x}=0,p_{y}=0) (large circle), (p_{x}=0,p_{y}=±2e A_{0}/π) (small circles). An analysis of this interference pattern allows us to determine the IR intensity to be I_{IR}=2.5×10^{13} W cm^{−2}, a value that agrees well with that estimated in our experimental conditions.
The complexity of the interference pattern obtained in Fig. 3c, combined with experimental statistical noise, prevent us from carrying out an analysis as in Fig. 2, except close to the p_{x} axis. This is equivalent to discussing the behaviour of the interference pattern in the region outlined by the white rectangle. In Fig. 4a, we compare these interference fringes with theoretical results obtained in argon for similar conditions as in the experiment (a train of Fouriertransformlimited 190attosecond pulses, centred at 27eV energy, and an IR intensity equal to I_{IR}=2.5×10^{13} W cm^{−2}). In this case, the momentum distribution measured is the incoherent sum of two contributions, originating from the 3p m=0 and the m=±1 states. Interestingly, because the corresponding groundstate wavefunctions have different symmetry properties (antisymmetric and symmetric relative to p_{y}=0, respectively), the interference fringes will be shifted by π along the p_{x} axis. Theoretical results obtained by separating the two contributions (Fig. 4b) clearly show this phase difference. Our experimental (and theoretical) results show that in this region, photoionization of argon is dominated by the contributions from the m=±1 states.
By increasing or decreasing the laser intensity, by varying the delay while taking the added accumulated phase due to the interaction with the IR field into account, and by using different polarization states for the XUV and IR light fields combined with a polarizationindependent detection technique^{21}, more information could be obtained in a larger region in momentum space. This opens exciting possibilities for investigating the continuum and groundstate electronic wavefunctions of complex, chemically relevant molecules and also for following the dynamics of timedependent superposition of states.
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Acknowledgements
This research was supported by Marie Curie IntraEuropean Fellowships (MEIFCT2004009268, MEIFCT2003500947), the Marie Curie Research Training Networks XTRA (MRTNCT2003505138) and PICNIC (HPRN200200183), the Integrated Initiative of Infrastructure LASERLABEUROPE (RII3CT2003506350) within the 6th European Community Framework Programme, the Knut and Alice Wallenberg Foundation, the Swedish Science Council and the National Science Foundation through grant PHY0401625. K.V. is on leave from the Department of Optics and Quantum Electronics, University of Szeged, Szeged, Hungary. The research of Y.N., F.L., M.K., J.K. and M.J.J.V. is part of the research program of the ‘Stichting voor Fundamenteel Onderzoek der Materie (FOM)’, which is financially supported by the ‘Nederlandse organisatie voor Wetenschappelijk Onderzoek (NWO)’. We thank T. Ruchon and M. Lewenstein for fruitful discussions.
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Remetter, T., Johnsson, P., Mauritsson, J. et al. Attosecond electron wave packet interferometry. Nature Phys 2, 323–326 (2006). https://doi.org/10.1038/nphys290
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DOI: https://doi.org/10.1038/nphys290
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